When \(Q=\widehat{P}\)  is analogous to \(P\)  (that is, no codimension two faces are truncated),
the push-pull polytope \(\Delta (P,Q)\)  coincides with the Newton polytope of a projective line
fibration \(Y=\mathbb {P}^1(E)\)  over the projective toric variety \(X\)  corresponding to the
polytope \(P\) .
We assume that \(E=\mathcal {O}\oplus \mathcal {L}\)  is split so \(Y\)  is also a toric variety.
In this case, there is a simple relation between the polytope \(Q\)  and the line bundle \(\mathcal {L}\) .
Namely, identify the Picard group of \(X\)  with the Grothendieck group of lattice polytopes analogous to \(P\) .
Denote by \([\mathcal {L}]\)  the (possibly virtual) polytope corresponding to \(\mathcal {L}\) .
Then \(Q=P+[\mathcal {L}]\) .
In terms of Khovanskii-Pukhlikov rings, this can rephrased as follows.
Under the isomorphism \(CH^*(X)\simeq R_P\)  the first Chern class \(c_1(\mathcal {L})\) 
gets mapped to \(\partial _{Q-P}\) .
In particular, the construction of Grossberg–Karshon cubes [1]} motivated by Demazure character formula (2) for Bott–Samelson varieties can also be reproduced in the spirit of (1) using convex-geometric push-pull operators.
We discuss this approach in more detail in Section REF .
